Abstract
An RD-space \({\mathcal X}\) is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in \({\mathcal X}\). In this paper, the authors first give several equivalent characterizations of RD-spaces and show that the definitions of spaces of test functions on \({\mathcal X}\) are independent of the choice of the regularity \({\epsilon\in (0,1)}\); as a result of this, the Besov and Triebel-Lizorkin spaces on \({\mathcal X}\) are also independent of the choice of the underlying distribution space. Then the authors characterize the norms of inhomogeneous Besov and Triebel-Lizorkin spaces by the norms of homogeneous Besov and Triebel-Lizorkin spaces together with the norm of local Hardy spaces in the sense of Goldberg. Also, the authors obtain the sharp locally integrability of elements in Besov and Triebel-Lizorkin spaces.
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Dachun Yang is supported by the National Natural Science Foundation (Grant No. 10871025) of China.
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Yang, D., Zhou, Y. New properties of Besov and Triebel-Lizorkin spaces on RD-spaces. manuscripta math. 134, 59–90 (2011). https://doi.org/10.1007/s00229-010-0384-y
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DOI: https://doi.org/10.1007/s00229-010-0384-y